Wikipedia defines it as the composition $g\circ f$. I'm confused about how to prove this is a zero morphism.
I'm also confused by when it is said that every category with a zero object is a category with zero morphisms given by the composition, do you have to show $g \circ f$ a left and right zero morphism, or do we have to show $C$ is a category with zero morphisms, which is different, I think?
To each object $X$ we assign a special morphism $f_X : X \rightarrow 0$ defined as the only such morphism, and another special morphisms $g_X : 0 \rightarrow X$ which is also the only such morphism.
For each pair of objects $X$ and $Y$, we define $0_{X,Y} = g_Y \circ f_X$.
Now just prove that $\varphi \circ 0_{X,Y} = 0_{X,Z}$ and $0_{Y,Z} \circ \psi = 0_{X,Z}$ for all morphisms $\varphi :Y \rightarrow Z$ and $\psi : X \rightarrow Y$.